Optimal. Leaf size=31 \[ \frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{6 d}-\frac{x}{12} \]
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Rubi [A] time = 0.0308961, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3783, 2657} \[ \frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{6 d}-\frac{x}{12} \]
Antiderivative was successfully verified.
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Rule 3783
Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{3+5 \sec (c+d x)} \, dx &=\frac{x}{3}-\frac{1}{3} \int \frac{1}{1+\frac{3}{5} \cos (c+d x)} \, dx\\ &=-\frac{x}{12}+\frac{5 \tan ^{-1}\left (\frac{\sin (c+d x)}{3+\cos (c+d x)}\right )}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0555576, size = 30, normalized size = 0.97 \[ \frac{2 (c+d x)+5 \tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 34, normalized size = 1.1 \begin{align*}{\frac{2}{3\,d}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{5}{6\,d}\arctan \left ({\frac{1}{2}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77395, size = 63, normalized size = 2.03 \begin{align*} \frac{4 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 5 \, \arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6259, size = 89, normalized size = 2.87 \begin{align*} \frac{4 \, d x + 5 \, \arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{5 \sec{\left (c + d x \right )} + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24765, size = 41, normalized size = 1.32 \begin{align*} -\frac{d x + c - 10 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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